167. The return of the phoenix universe
Save This Post as PDFThis is a guest blog post by Jean-Luc Lehners from Princeton University. Dmitry.
In a recent paper entitled “Dark energy and the return of the phoenix universe” that I wrote with Paul Steinhardt, we analyzed the fate and global structure of the two-scalar-field cyclic universe. (In the cyclic universe the big bang is modeled by the regular collision of two end-of-the-world branes, see section 5 of my review.) From the point of view of string theory, this is a natural model to consider, as there always are two scalar fields in braneworld models: one parameterizing the distance between the two branes, and one determining the volume of the internal space. Of course there could be many more such moduli, but this is the minimal model. From the point of view of cosmology, this model constitutes a rather nice setup, as it allows the density perturbations to arise via the entropic mechanism, in which nearly scale-invariant entropy perturbations are produced first, and these are then converted into adiabatic perturbations just before the big crunch.
The entropic mechanism is the best-understood mechanism for producing nearly scale-invariant, adiabatic density perturbations in ekpyrotic/cyclic models, but it comes at a price: it renders the background trajectory unstable. This is because the two scalar fields are assumed to have steep, negative potentials (corresponding to an attractive force between the branes), and the background trajectory must evolve along a ridge of the potential. Only if the trajectory remains on the ridge for sufficiently long will the universe become highly flat, homogeneous and isotropic, and only then will the density perturbations have the required amplitude. If the background trajectory falls off the ridge too early, then the universe does not become flat and the result will be a universe that collapses in a mixmaster crunch and/or recollapses rapidly. Thus it is not clear at first sight whether cycling is actually viable in the presence of this instability. What to do? Well, as we found out, there’s really nothing we have to do. The model takes care of itself! The reason is that, although most trajectories starting near the ridge will fall off prematurely, with the consequence that the corresponding spacetime regions collapse and stop cycling, the small subset of trajectories that are exponentially close to the ideal cyclic trajectory will result in a flat universe that grows by an exponential amount over the course of the next cycle. Hence we have a competition of two exponentials, and it turns out that if the dark energy phase lasts for at least 60 e-folds, then the absolute size of the flat regions grows from cycle to cycle. Given what we currently know about dark energy, 60 e-folds (or about 600 billion years of accelerated expansion at the current rate) do not seem like a strong requirement.
During each cycle, the instability causes most trajectories to fall off prematurely. Hence, within each flat, habitable region, most of space gets lost in the next big crunch, i.e. the universe turns to ashes. However, a small seed will grow to become the new habitable universe. Because of this analogy, we found the name “phoenix universe” particularly fitting – see also the figure for a sketch of the corresponding global structure. In fact, the name “phoenix” seems even more appropriate here than in the original phoenix universe considered by Georges Lemaitre in the 1930s. Lemaitre’s phoenix was a closed, positively curved universe that periodically recollapsed. Moreover, it was implicitly assumed that the entire universe would bounce back at the big crunch. Also, Lemaitre’s phoenix had a positive curvature and was a decelerating universe, whereas here the braneworld structure allows us to have a flat and accelerating universe which nevertheless evolves into a contracting phase and a big crunch. As is clear from the above discussion, dark energy plays an absolutely central role here in stabilizing the universe.
Let me also mention another way of phrasing the central argument, which involves thinking in terms of field gradients: in order for the flat regions to grow from cycle to cycle, in a given volume of space (which we imagine to be part of a flat region, and at a specified epoch in each cycle) the field gradient has to decrease from cycle to cycle. Now, the unstable ekpyrotic potential causes the field gradient to increase by about 120 e-folds, whereas the standard radiation and matter expansion eras decrease it by 60 e-folds. Hence we need an additional 60 e-folds or more of dark energy dominated expansion to achieve an overall smoothing of the field configuration from cycle to cycle.
Apart from the special global structure described above, are there other notable consequences of this model? Well, the most important observational signature is the level of non-Gaussianity expected in the CMB temperature fluctuations. Cyclic models in which the perturbations arise via the entropic mechanism predict the non-linearity parameter 
to be of order a few 10s. Hence we would reasonably expect Planck or a similarly sensitive experiment to detect it. Also, as discussed above, we can expect spatial variations in dark energy to decrease from cycle to cycle. So, even though it seems unlikely (though not impossible, see the paper for details) that the magnitude of the spatial variation of dark energy would be at an observationally accessible level today, nevertheless, conceptually, spatial variations of dark energy play the role of an inter-cyclic clock. This is just one example of a quantity that evolves over timescales much longer than the time since the last big bang. Such timescales may turn out to be very interesting regarding the evolution of the universe, as they may resolve otherwise puzzling issues of apparent fine-tuning.
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